A Slope Problem Involving a Billiards Table
Date: 10/19/2004 at 21:09:29 From: Christie Subject: The Billiard Problem Suppose that a square billard table has corners at (0,0),(1,0),(1,1), (0,1). A ball leaves the origin along a line with slope s. If the ball reaches a corner, it stops or falls in. Whenever it hits the side of the table not at a corner, it continues to travel on the table, but the slope on the line is multiplied by -1. Does the ball reach a corner? The solution to the problem is when s = 3/5. I do not understand why.
Date: 10/20/2004 at 11:41:00 From: Doctor Douglas Subject: Re: The Billiard Problem Hi, Christie. Thanks for writing to the Math Forum. I think what is meant is that if the trajectory passes through one of the four corners, then the ball stops. You can analyze this problem by considering the grid of all points (x,y) where x and y are nonnegative integers. If the initial trajectory from the origin hits one of the points in the entire grid, then the ball will stop there (if it hasn't already stopped). The case of initial slope s = 3/5 will certainly hit the grid point at (x,y) = (5,3), which is point A in the figure below. You can verify that it doesn't hit any earlier. y | | | | | | | 3 - . - . - . - . - A - . - | | | | | | | 2 - . - . - . - . - . - . - | | | K | | | | 1 - . - . - . - . - . - . - | | | | | | | O - 1 - 2 - 3 - 4 - 5 - 6 - x The line from O to A passes through a number of lines (I calculate six). If it passes through a vertical line, the ball bounces on one of the left/right walls. If it passes through a horizontal line, it bounces on either the top or bottom walls. By drawing the line from O to A, you can track the entire history of how many times the ball bounces and exactly where it goes. For example, you can convince yourself that the ball passes through the exact middle of the table at point K, halfway through its trajectory. Many other trajectories lead to the ball hitting one of the corners. In fact, any slope s = p/q that is rational will cause it to hit a corner [at (q,p), if not earlier]. Slopes s that are irrational (such as pi, or sqrt(2) will allow the ball to bounce around the table forever. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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